Question

A Card is drawn from a pack of $52$ cards, so that each card is equally likely to be drawn from a pack. ${}^{\prime }{A}^{\prime }$ is the event that the card drawn is of 'Spade' and ${}^{\prime }{B}^{\prime }$ is the event, where the card drawn is an 'ace'. Explain whether the two events are independent.

Answer 1

$\to$ If $A$ and $B$ are independent events, $P(A\cap B)$ should be equal to $P\left(A\right)\times P\left(B\right)$

Here, $P\left(A\right)=\frac{13}{52}=\frac{1}{4}$

$P\left(B\right)=\frac{4}{52}=\frac{1}{13}$

and $P(A\cap B)=\frac{1}{52}$

$\Rightarrow P(A\cap B)=P\left(A\right).P\left(B\right)$